A conceptual circuit for manually setting the speed of a dc electric motor is shown in Fig. 3.31a. The resistors R_1 and R_1 are inside a component called a potentiometer, or pot, which is nothing more than an adjustable resistor, for example, a volume control. Turning the knob changes the ratio \alpha=R_{2} /\left(R_{1}+R_{2}\right), but the total resistance, R_{\text {pot }}=R_{1}+R_{2}, is unchanged. In this way the pot forms a voltage divider that sets the voltage V_{\text {speed }}. The power amplifier output, V_{M}, is four times V_{\text {speed }}. Power amplifiers can output the high currents needed to drive the motor. Finally, the dc motor speed is proportional to V_{M}; that is, the speed in rpm is some constant k times V. Without knowing the details of the power amplifier, can we analyze this system? In particular, can we develop a relationship between rpm and α ?